Location: Roger Stevens LT 16
Title: Building generalized indiscernibles in nonelementary classes
NOTE location change: Roger Stevens LT 16
Generalized indiscernibles can be built in first-order theories by generalizing the combinatorial Ramsey’s Theorem to classes with more structure, which is an active area of study. Trying to do the same for infinitely theories (in the guise of Abstract Elementary Classes) requires generalizing the Erdos-Rado Theorem instead. We discuss various results about generalizations of the Erdos-Rado Theorem and techniques (including large cardinals and forcing) to build generalized indiscernibles.
Location: Social Sciences SR (14.33)
Title: Generalised measurability and bilinear forms
NOTE location change: Social Sciences SR (14.33)
In this talk I will go over measurable and generalised measurable structures, giving examples and non-examples. I will then go on to consider the two sorted structure $(V,F,β)$ where $V$ is an infinite dimensional vector space over $F$ an infinite field, and $β$ a bilinear form on this vector space. In particular I will consider the interaction of different notions of independence when this structure is pseudo finite. I will finish with some questions around generalised measurable structures.
Location: MALL 1
Title: Model Theory and Combinatorics or (The Unexpected Virtue of Tameness)
Hrushovski once called model theory the "geography of tame mathematics"; a quote which may make no sense if you're not a model theorist (or if you are, in fact, a geographer). I will try to explain this point of view, by discussing how model theorists draw "dividing lines" separating the tame theories from the wild ones and illustrating this through examples from combinatorics.
The aim of this talk is to provide a gentle introduction to some recent developments in the area. Perhaps ambitiously, I will try to (briefly and informally) discuss three very different topics: (i) Szemerédi Regularity in stable (due to Malliaris-Shelah), NIP (due to Fox, Pach and Suk), and distal structures (due to Chernikov and Starchenko); (ii) Algorithmic tameness in hereditary classes of relational structures (joint work with S. Braunfeld, A. Dawar and I. Eleftheriadis); and (iii) Zarankiewicz's problem in semilinear (due to Basit, Chernikov, Starchenko, Tao and Tran) and semibounded (joint work with P. Eleftheriou) o-minimal structures.
Location: MALL
Title: Pseudo-finite semigroups and diameter
A semigroup $S$ is said to be (right) pseudo-finite if the universal right congruence $S \times S$ can be generated by a finite set $U$ of pairs of elements of $S$ and there is a bound on the length of derivations for an arbitrary pair as a consequence of those in $U$ . The diameter of a pseudo-finite semigroup is the smallest such bound taken over all finite generating sets.
The notion of being pseudo-finite was introduced by White in the language of ancestry, motivated by a conjecture of Dales and Zelazko for Banach algebras. The property also arises from several other sources.
Without assuming any prior knowledge, this talk investigates the somewhat unpredictable notion of pseudo-finiteness. Some well-known uncountable semigroups have diameter $1$; on the other hand, a pseudo-finite group is forced to be finite. Actions, presentations, Rees matrix constructions and some good old-fashioned semigroup tools all play a part.
This research sits in the wider framework of a study of finitary conditions for semigroups.
Location: MALL
Title: Combining Manin-Mumford and weak Zilber-Pink
I will introduce some classical notions and problems in Diophantine geometry, including the Manin-Mumford and Zilber-Pink conjectures, and explain how model-theoretic tools are used to approach them. I will then talk about one of my recent theorems establishing a new partial result towards Zilber-Pink by combining Manin-Mumford and a weak version of Zilber-Pink (both are theorems). I am going to start with very basic things, give quite a few examples and define/explain all concepts that I am going to use, so I hope that most of the talk will be accessible to a wide range of people including those who have not heard about Diophantine geometry before.
Location: MALL
Title: What makes the continuum $\aleph_2$
While historically the question has been whether the Continuum Hypothesis is true or false, determining the relationship between the continuum and $\aleph_2$ (the second uncountable cardinal) is arguably a much deeper and more interesting mathematical problem. I will lay out a philosophical and mathematical argument for why $\aleph_2$ is the right value for the continuum.
Location: MALL
Title: What, why, and how are Forcing Axioms?
Forcing axioms are set-theoretic axioms which postulate the existence of "somewhat generic objects" in the universe. The goal of this talk is to give a fairly accessible explanation of this sentence, to understand some of the consequences of forcing axioms, and in what sense some forcing axioms are stronger than others
Location: MALL
Title: Ordinal arithmetic and subgroups of Thompson's group
The class of finitely generated groups embeddable into Richard Thompson's group $F$ is both restrictive and rich at the same time. We show that there is a family of groups within this class which is pre-well-ordered in type $\epsilon_0$ by the embeddability relation. Moreover, the operations of addition and multiplication on the ordinals translate into natural group-theoretic operations—direct sum and a type of permutational wreath product. This talk will give a description of this correspondence. This is joint work with Collin Bleak and Matt Brin.
Location: MALL
Title: A proof theorist's job, part II: into harmonious systems
This talk will be about ordinal analysis, a technique at the core of a proof theorist's toolchain. Using Tait calculus as the formal proof system for arithmetic, we will replicate Gentzen's famous consistency proof as the seemingly useless statement "enough of set theory ⊢ Con(PA)". Hopefully, I will convince the audience that the left-hand side of the turnstile can be weakened to a theory of transfinite induction formalised in primitive recursive arithmetic, and this proof thus measures the proof-theoretic strength of a formal theory using the linear ordering in a given ordinal representation system.
Location: MALL
Title: Indestructibility and $C^{(n)}$-supercompact cardinals
In the 70's Laver showed that a supercompact cardinal $\kappa$ may be made indestructible by a suitable class of forcings—namely, after a preparatory forcing, the supercompactness of $\kappa$ will not be destroyed by any further $<\kappa$-directed closed forcing. Many indestructibility results have since been written, as well as those demonstrating the impossibility of indestructibility (or even preservation) of many large cardinals. In this talk we will consider the case of $C^{(n)}$-supercompact cardinals—a stronger and more slippery variant of supercompact cardinals—and how they can be made indestructible for $n\leq 2$.
